The excision theorem for digital Khalimsky spaces

2015

In this paper we introduce the digital singular homology groups of the digital spaces topologized by the Khalimsky topology by constructing the digital standard n-simplexes. Then we'll compute the digital singular homolo gy groups of some basic digital spaces up to the dimension 2 and investigate that the digital singul ar homology theory for the digital spaces is a functor from the category KDTC of KD-topological category to the category Ab of abelian groups.

TURKISH JOURNAL OF MATHEMATICS, 2020

In this paper, we develop homology groups for digital images based on cubical singular homology theory for topological spaces. Using this homology, we present digital Hurewicz theorem for the fundamental group of digital images. We also show that the homology functors developed in this paper satisfy properties that resemble the Eilenberg-Steenrod axioms of homology theory, in particular, the homotopy and the excision axioms. We finally define axioms of digital homology theory. Keywords digital topology • digital homology theory • digital Hurewicz theorem • cubical singular homology for digital images • digital excision Mathematics Subject Classification (2010) 55N35 • 68U05 • 68R10 • 68U10

Topology Proceedings, vol. 61, 2023

Let \scrF be any collection of subspaces of a topological space and let \scrD ⊆ \scrF . We say \scrD is homology-simple in \scrF if the inclusion map of \bigcup (\scrF \setminus \scrD ) in \bigcup \scrF induces homology group isomorphisms in all dimensions, and say \scrD is hereditarily homology-simple in \scrF if every subcollection of \scrD is homology-simple in \scrF. Thinning algorithms are used in image processing to simplify binary images. An nD binary image may be regarded as a representation of a finite collection of closed n-cubes that are grid cells of an nD Cartesian grid. Then the goal of thinning is to reduce such a collection of grid cells to a subcollection that is a "thin skeleton'' of the collection. Thinning algorithms are commonly designed in such a way that, if \scrF_in is the original collection of grid cells and \scrF_skel the resulting skeleton, then \scrF_in \setminus \scrF_skel is homology-simple in \scrF_in. We call this the homology preservation condition. For n \in \{ 2, 3, 4\} , a theorem of Bertrand and Couprie implies a local characterization of the hereditarily homology-simple subcollections of any finite collection \scrF of nD Cartesian grid cells. The theorem and the implied characterization of hereditary homology- simpleness can be used to design good parallel thinning algorithms that automatically satisfy the homology preservation condition, and can also be useful for verifying that a proposed parallel thinning algorithm satisfies that condition. Bertrand and Couprie's work makes no explicit use of homology. It is based on collapsing of complexes and depends on \scrF being part of a cubical complex of dimension \leq 4. This paper shows how their theorem can be restated using homology, and shows that the restated theorem is valid under much weaker hypotheses than were assumed in the original theorem. For example, it is valid whenever \scrF is a finite collection of acyclic polyhedra in R^n whose nonempty intersections are acyclic, and whenever \scrF is a finite collection of acyclic open sets (of any topological space) whose nonempty intersections are acyclic. We also deduce a characterization of hereditary homology-simpleness in the case where \scrF is a finite collection of singleton subspaces of Khalimsky n-space.

In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate Eilenberg-Steenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for simplicial homology doesn't be hold in digital images. Moreover we show that the Hurewicz Theorem need not be hold in digital images. Copyright © AJCTA, all rights reserved.

Discrete & Computational Geometry, 2021

Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCES, 2016

In this paper, we study certain properties of digital H-spaces. We prove that a digital image that has the same digital homotopy type with any digital H-space is also a digital H-space. We show that the digital fundamental group of a digital H-space is abelian. We give examples that are related to a digital homotopy associative H-space and a κ-contractible digital H-space. Several important applications of digital H-spaces are given in computer vision and image processing. Finally, we deal with the importance of digital H-space in digital topology and image processing. We conclude that any κ-contractible digital image is a digital H-space.

Discrete Applied Mathematics, 2003

The main contribution of this paper is a new "extrinsic" digital fundamental group that can be readily generalized to define higher homotopy groups for arbitrary digital spaces. We show that the digital fundamental group of a digital object is naturally isomorphic to the fundamental group of its continuous analogue. In addition, we state a digital version of the Seifert-Van Kampen theorem.

In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an n-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a 'fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.

Discrete Applied Mathematics, 2004

Mediterranean Journal of Mathematics, 2011

In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called Snake space. In the sequel we introduced the functor SC(−, −) defined on the category of all spaces with base points and continuous mappings. For the circle S 1 , the space SC(S 1 , * ) is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces SC(Z, * ) for any path-connected compact spaces Z.